Step by step
The question
Question: the line A T touches the circle at A, the angle between it and the chord A B is 68°, and angle A B C is 57°: find angle B A C.
1 Use the rule
The angle between the tangent and the chord A B is 68°.
The alternate segment theorem: it equals the angle at C, on the other side of the chord, in the alternate segment.
So angle A C B is 68°.
angle ACB = 68°
Careful!
Not the angle at B, next to the tangent angle.
The equal angle is across the chord, at C.
Careful: the angle in the OTHER segment, across the chord
2 Use the angle fact
The three angles of triangle A B C add up to 180°.
180 minus 68 minus 57 is 55°.
angle BAC = 180° − 68° − 57° = 55°
3 Check
Let's check: on the straight tangent line at A, the angle on the other side is 57°, by the same rule.
68 plus 55 plus 57 is 180°: a straight line.
So it is right!
on the tangent at A: 68° + 55° + 57° = 180°
Let's look back.
First: use the rule.
Then: use the angle fact.
Last: check.
Answer
angle BAC = 180° − 68° − 57° = 55°